有梯形一腰中点时,常过此中点作另一腰的平行线,把梯形转化成平行四边形 例:已知,如图,梯形ABCD中,AD∥BC,E为CD中点,EF⊥AB于F,求证:S梯形ABCD = EF·AB
证明:过E作MN∥AB,交AD的延长线于M,交BC于N,则四边形ABNM为平行四边形 ∵EF⊥AB ∴S□ABNM = AB·EF ∵AD∥BC ∴∠M =∠MNC 又∵DE = CE ∠1 =∠2 ∴△CEN≌△DEM ∴S△CEN = S△DEM ∴S梯形ABCD = S五边形ABNED+S△CEN = S五边形ABNED+S△DEM = S梯形ABCD = EF·AB
有梯形一腰中点时,也常把一底的端点与中点连结并延长与另一底的延长线相交,把梯形转换成三角形 例:已知,如图,直角梯形ABCD中,AD∥BC,AB⊥AD于A,DE = EC = BC,求证:∠AEC = 3∠DAE
证明:连结BE并延长交AD的延长线于N ∵AD∥BC ∴∠3 =∠N 又∵∠1 =∠2 ED = EC ∴△DEN≌△CEB ∴BE = EN DN = BC ∵AB⊥AD ∴AE = EN = BE ∴∠N =∠DAE ∴∠AEB =∠N+∠DAE = 2∠DAE ∵DE = BC BC = DN ∴DE = DN ∴∠N =∠1 ∵∠1 =∠2 ∠N =∠DAE ∴∠2 =∠DAE ∴∠AEB+∠2 = 2∠DAE+∠DAE 即∠AEC = 3∠DAE
梯形有底的中点时,常过中点做两腰的平行线 例:已知,如图,梯形ABCD中,AD∥BC,AD<BC,E、F分别是AD、BC的中点,且EF⊥BC,求证:∠B =∠C
证明:过E作EM∥AB, EN∥CD,交BC于M、N,则得□ABME,□NCDE ∴AE = BM,AB∥= EM,DE = CN,CD = NE ∵AE = DE ∴BM = CN 又∵BF = CF ∴FM = FN 又∵EF⊥BC ∴EM = EN ∴∠1 =∠2 ∵AB∥EM, CD∥EN ∴∠1 =∠B ∠2 =∠C ∴∠B = ∠C
任意四边形的对角线互相垂直时,它们的面积都等于对角线乘积的一半 例:已知,如图,梯形ABCD中,AD∥BC,AC与BD交于O,且AC⊥BD,AC = 4,BD = 3.4,求梯形ABCD的面积.
解:∵AC⊥BD ∴S△ABD =1/2AO·BD S△BCD=1/2CO·BD ∴S梯形ABCD = S△ABD +S△BCD=1/2AO·BD+1/2CO·BD=1/2(AO+CO)·BD 即S梯形ABCD=1/2AC·BD =1/2×4×3.4=6.8 答:梯形ABCD面积为6.8.
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